Elements of South-Indian Palæography, from the Fourth to the Seventeenth Century A.D. c6: Chapter III. The South-Indian Numerals

Elements of South-Indian Palæography, from the Fourth to the Seventeenth Century A.D. c6

Language: English

c6.14 — 4 parts
c6.s1 § 1. The Modification of the 'Cave' Numeral Figures Found in the Veṅgi and Cēra Inscriptions — 5 parts
c6.s2 § 2. The Tamil Numeral Figures — 8 parts
c6.s3 § 3. The Telugu-Canarese Numeral Figures — 6 parts
c6.s4 § 4. The Javanese Numeral Figures — 4 parts
c6.s5 Note: The Different Methods of Marking Dates Used in South-India — 4 parts

Chapter III. The South-Indian Numerals

THE history of the numerals used in India is of the last importance, as on it, partly, depends the solution of a very important question—the origin of the European decimal systems of notation by which the value of the numbers depends on position and which also involves the use of the cipher. The facts furnished by the S. Indian inscriptions unfortunately are of little more value than to throw doubts on the speculative conclusions arrived at by Woepcke originally[^1), but which are now commonly asserted in popular manuals[^2). These are: that the early Indian numeral signs and ciphers are derived from the initial letters of the words denoting the same; that these numeral figures were brought to Europe by two distinct courses—firstly, about the early centuries of our era by Neo-Pythagoreans through the intercourse between Alexandria and India; and secondly, by the Arabs, who adopted them about the ninth century[^3). The last proposition is the only one of the three which rests on any historical evidence; the rest are inferences drawn by Woepcke with some probability, and have been so far accepted by the most eminent Indianists[^4). Whether the inscriptions that have been discovered

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[^1): Woepcke, “Mémoire sur la propagation des chiffres Indiens” (separate impression, 1863) pp. 2-3. The author mentions the imperfect evidence, and then asks if all attempt to draw conclusions must be abandoned. His own opinion he states as follows: “Je ne le pense pas, pourvu qu’en tâchant de construire un ensemble, on fasse consciencieusement connaître les parties conjecturales pour les distinguer d’avec les parties certaines, et pourvu que l’on ne présente les explications hypothétiques auxquelles on est obligé de recourir que comme la résultante la plus probable des faits connus dans le moment; pourvu en fin que l’on soit toujours prêt à modifier ses conclusions dans le cas où la découverte de documents nouveaux en rendrait la nécessité évidente.” It appears to me that the explanation of the Cave numerals, and the ascertainment of the complete series of units, as well (as I shall show) that these numerals were used over greater part of S. India, now warrant a different conclusion to that of Woepcke as regards the origin of the current figures. [^2): A. Braun (Die Ergebnisse der Sprachwissenschaft, p. 26.) says: Dass einige dieser Ziffern eine grosse Aehnlichkeit mit den unsrigen haben, sieht man sofort. In der That verdienen unsere Zahlzeichen es eigentlich auch nicht, arabische genannt zu werden, denn sie stammen ursprünglich aus Indien; die Araber waren nur die Ueberbringer, nicht die Erfinder derselben”. [^3): For the first proposition see pp. 44-52 of Woepcke’s “Mémoire”; for the second, pp. 123-6; as regards the third, the Indian Embassy to Al-Mançūr was in 773 A. D. It seems very probable, however, that the chief scientific information that the earlier Arabs got from India came entirely through Persia. *Cfr.* Dr. Haas’s valuable Paper on Indian Medicine, in the Z. d. D. M. G. vol. xxix., and the quotation from the Dīnkart in Haug’s “Essay on Pahlavī,” p. 146. [^4): Max Müller, “Sanskrit Grammar” p. 9 (2nd ed.); “Chips from a German Workshop,” ii., p. 295. Also by Profr. Benfey in his “Geschichte d. Sprachwissenschaft”, p. 802.

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since these conclusions were arrived at, as well as some facts as yet unnoticed, do or do not support them, is now a matter for serious enquiry.

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The earliest known examples of Indian numeral figures occur in the Açoka inscriptions which have already been often mentioned, and which belong to the middle of the third century B. C. In the Kapurdigiri inscription the number ‘four’ is expressed by four upright lines, thus ||||¹. Later inscriptions in the same character furnish other examples; the most important is one from Taxila, which is of the first century B. C. and in which the number 78 is expressed by 3× 20 + 1× 10 + 2× 4; the figures for 20, 10 and 4 being distinct signs. The figures for ‘four’ in these two inscriptions (|||| and +) show a considerable development between the third and first centuries B. C. It is, therefore, certain that the method of denoting numerals, which prevailed in the early centuries B. C. in the Panjab and Ariana, began with the use of strokes equal to the number to be expressed, and that this primitive system had, by no means, become perfect in the first century B. C.

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¹ The late illustrious scholar who deciphered this inscription (Mr. E. Norris) told me that this gave him the clue by which he recognized it as an Açoka edict, and was thus able to decipher it.

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The Southern Açoka inscriptions, which, as I have already said, are alone of importance for South-Indian palæography, do not contain any numeral signs except the Khalsi inscription which has × for ‘four,’ and those recently discovered at Rūpnāth and Sahasrām which have figures for 256, but which are, probably, somewhat more recent. But there are inscriptions from Mathura, which are in nearly the same character, belonging to the first or second century A. D., *probably*, which show a well-developed system entirely distinct from that which is found in the Arianic inscription of Taxila of about the same date. In this the first three numerals are expressed by one, two and three horizontal strokes, the rest (four, etc.) have distinct figures; and there is a distinct figure for each of the orders of numbers (ten, twenty, etc.) up to one hundred which has, as well as one thousand, a sign to itself. The intermediate units are expressed by simply adding their signs; for example, ‘twenty-five’ is expressed by the sign for ‘twenty’, followed by that for ‘five’. There is not the least trace of the use of the cipher in this system. It is obviously an independent and ingenious development of much the same elements as were used in the Arianic system, but far more perfect. It is quite impossible to derive these signs from the initial letters of the words for the numbers, as they bear no resemblance at all to the Southern Açoka letters which begin the corresponding words, nor excepting the signs for ‘eight’ and ‘nine’ do they bear any resemblance to the same letters in the Kapurdigiri character; and the likeness in both these cases is very superficial. This system of numerals was used in the Cave inscriptions of Western India, and in many other parts of India during several centuries; owing to the combined researches of several scholars this system is now fully understood. The latest inscription in N. India appears to be of the fourth century¹), but nearly the same numerals occur in inscriptions of the early Vengi dynasty of Kalinga which must be referred to the fourth and fifth century, and the sign for 'ten' occurs in a Cera inscription *d.* 466 A. D. The system of numeral figures, still used by the Tamiḻ people, forms a step in advance, the distinct signs for 'ten', 'hundred' and 'thousand' only being preserved, and those for 'twenty' up to 'ninety' being discarded. Apart from this still existing system, there is no evidence as to the use of these 'Cave numerals', as they are usually termed, after the fifth century, for inscriptions with dates in figures are, as yet, wanting from that time till about the tenth century in Northern India, and till about the year 1000 A. D. in S. India²). At these dates, we find, in the respective countries, the exclusive use of numeral figures with a value according to position and the cipher; and the figures have much the same forms as are now current³), and which so closely resemble the Gobar numerals, also in use and with the same value according to position in Europe also about the eleventh century. Though it has often been asserted that the modern, or Nāgarī, numerals are mere abbreviations of the initial letters of the words denoting the corresponding numbers⁴), I think that a comparison of the later forms of the Cave numerals with them, will render it perfectly clear, that, this is not the case, but that, all the indigenous numerals used in the various parts of India are simply derived from the Cave numerals which are not, as I have already shown, derived from the initial letters of the corresponding words. This derivation is also the only one which satisfactorily explains the forms of the numerals used in the North as well as in the South of India⁵) and Java.

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¹ The Kaira plates. See Prinsep's "Essays" by Thomas, I. p. 257.

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² A Kawi inscription of the 9th century, however, bears a date (763 ç.) in figures. See pl. ii. of Dr. Cohen Stuart's "Kawi Oorkonden." About 880 A. D. the cipher (0) was in common use in Java (do: pl. ix., xi., xiv., xv.).

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³ According to Albīrūnī (Reinaud, "Mémoire", p. 299) the numeral figures aṅka used in different parts of India in the 11th century varied in form, but value by position was generally known.

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⁴ Woepcke, "Mémoire" pp. 44-53.

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⁵ See pl. xxiii.

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It therefore appears that, neglecting all possibilities, in favour of which evidence does not exist, (such as the simultaneous existence of the more modern system of notation with the older in the fifth century A. D. or even earlier), the only possible conclusion is that, the great improvement of using numerals with a value according to position, and consequently the use of the cipher, first occurred in Central India about 500 A. D. Now though the inscriptions fail us as yet for this period, the acuteness of Woepcke detected some evidence in the works of the astronomers who lived in India during the fifth and following centuries. These are: Āryabhaṭa who himself tells us that he was born in 476 A. D.¹, Varāha Mihira who died in 587 A. D.², Brahmagupta who lived about 600, and Bhaṭṭotpala who lived about 1000 also of our era. All these writers composed their treatises in metre, and to suit the exigencies of the strict limits thus imposed on them, the three last were obliged to express the terms of their calculations by words, and these not the usual ones, but by symbolical words denoting natural objects, and in a conventional way, (as here used) also numerals. This peculiar system (which will be fully explained further on in this chapter) implies value by position, and also has words which express *indirectly* the cipher³. This same system is also used in the Sūryasiddhānta which is of very uncertain date in its actual form.

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It is thus perfectly clear that the Indians knew of numerals with a value according to position in the sixth century A. D., but the system of Āryabhaṭa which is totally different to the one described, *appears* to render improbable the assumption that he also about 500 A. D. found this system in common use, though he was acquainted with it⁴. He himself uses the successive vowels of the Sanskrit alphabet to express place, and thus his system agrees in principle with the Tamil notation; **a**, **ā** and **i** corresponding in value with the Tamil signs for ‘ten’, ‘hundred’ and ‘thousand.’ Woepcke, however, considers that Āryabhaṭa invented this notation to suit his style of composition in verse, and that the system of notation by words with value according to position was ‘probably anterior to Āryabhaṭa’⁵. If it had been then in common use, would not Āryabhaṭa have used it? Beyond the end of the fifth century there is, therefore, nothing to indicate the use of the cipher; for the high orders of numerals (equivalent to billions, trillions, etc.) first noticed by Profr. Weber⁶ do not necessarily imply anything of the kind. An illustrious French mathematician and a more eminent French philosopher have shown that this invention may have occurred (as was the case) in the middle ages of Europe spontaneously¹);

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it must also have occurred independently in India, but as the facts stand at present, it is impossible to connect India and Europe in the transmission of this particular invention from the first country to the other. As it is not proved to have been known in India before about 500 A. D., it is impossible to see how it can have been transmitted from thence to Europe before the rise of the Arabs, for direct communication ceased about the fourth century A. D.²), and in Europe, at all events, the little intellectual activity that was displayed ran in entirely different courses during the sixth, seventh, and eighth centuries. Nor is there the least trace of the use of such numerals during this period. The Arabic numerals now in use certainly came from India, but numerals with value according to position and the cipher were already in use in Europe (by the Neo-Pythagoreans) before they were adopted³). If the derivation of the numeral figures from the initial letters of the Sanskrit words denoting the respective numbers be given up, there is nothing left to show where the figures were first used, by the Pythagoreans in Europe or the astronomers in India. The assumption that the last was the case, but which (as I have already said) an examination of the earliest forms of the numerals preserved in inscriptions will prove to be impossible, is the foundation of the theory that Europe is indebted to India in this respect;

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in fact Woepcke chiefly relies on it⁴), but this assumption was made by J. Prinsep about 1838, or long before the Cave numerals were explained. The resemblance between the Neo-Pythagorean numerals in their cursive form on the one hand, and the India Cave numerals and the forms derived from them on the other, is too striking not to be noticed; but this fact does not warrant a presumption that one is borrowed from the other; more probably, both are from a common source. The question of what might have been the common original of the Neo-Pythagorean and Cave numerals is one for the decision of which evidence must yet be discovered, but a provisional hypothesis may be allowed.

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¹ See p. 58, (iii. 10), of Profr. Kern’s admirable edition.

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² Bombay Journal, viii., p. 241.

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³ It must be remarked that these words all mean ‘*blank*’, ‘*vacancy*’ or ‘*sky*’, and that there is nothing to show that there was a distinct *mark* or *figure* for the cipher; thus this Indian notation by words exactly corresponds with the system of the *abacus*. Woepcke wrongly translates two of these words (çūnya and kha) by ‘*le point*’ (p. 103), and there is therefore nothing in these astronomical treatises to show that the figure cipher was used in India even in the sixth century A. D. The Indian *abacus* was by using heaps of cowries for the numbers, the number of these shells being equal to that of the number to be expressed, the cipher being a blank space. Thus ∴| |∴=303; ∴|∴=33. Warren “Kala Sankalita”, p. 334) mentions a counter as used to express the cipher, but I have never found this to be done.

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⁴ I owe this correction to M. Barth (“Revue Critique”, 28 Août, 1875, p. 132). May not Āryabhaṭa be the discoverer of the decimal notation in India?

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⁵ u. s. p. 117 *n*. “Il ne faudrait pas conclure de l’existence d’une notation alphabétique inventée par Āryabhata que cette invention est nécessairement antérieure à celle des chiffres. Āryabhata, qui écrivait aussi en vers, avait besoin d’une notation qui se laissait mettre en çlōkas, et trouvait peut-être que la méthode des mots symboliques, très probablement antérieure à Āryabhata, manquait de brièveté et de précision.” Āryabhata (so the MSS. have his name) wrote in Ārya metre, and words would suit him better than letters; the fact remains that he did *not use* value by position.

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⁶ Z. d. D. M. G. xv., pp. 132 ffg.

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¹ Chasles who is supported by Comte. The last says (“Cours de Philosophie Positive”, v. p. 326 *note*): “Personne n’ignore ni l’heureuse innovation réalisée au moyen âge, dans les notations numériques, ni la part incontestable de l’influence catholique à cet important progrès de l’arithmétique. Un géomètre distingué, qui s’occupe, avec autant de succès que de modestie de la véritable histoire mathématique (M. Chasles), a très-utilement confirmé, dans ces derniers temps par une sage discussion spéciale, au sujet de ce mémorable perfectionnement, l’aperçu rationnel que devait naturellement inspirer la saine théorie, du développement humain, en prouvant qu’on y doit voir surtout, non une importation de l’Inde par les Arabes, mais un simple résultat spontané du mouvement scientifique antérieur, dont on peut suivre aisément la tendance graduelle vers une telle issue par des modifications successives, en partant des notations primitives d’Archimède et des astronomes grecs”. The *abacus* of the ancients was so near the modern system of numeration, that they would have but little felt the want of it. Martin (“Histoire de l’ Arithmétique”) had already in 1857 traced our decimal notation to a natural transformation of the *abacus* about 1100-1130 A. D., and this has now been rendered beyond doubt by Narducci’s most important discovery that the Boethian numerals with value by position were already in use at the end of the 12th century, an earlier date than that at which the Arabic numerals were so employed in Europe. See his Essay: “Intorno ad un Manuscritto della Bibliotheca Alexandrina continente gli Apici di Boezio senz’ Abaco e con Valore di Posizione.” Rome, 1877.

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² Reinaud “Relations politiques et commerciales de l’empire Romain”, p. 265-9. Woepcke (“Mémoire” p. 67) allows that if the invention came from India, it must have been transported thence: “dans les premiers siècles de notre ère.” Thus his hypothesis is no longer tenable, for the Gobar figures do not resemble the early forms.

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³ “Mémoire”, p. 194. This has, however, been doubted, but without sufficient reason, as now fully appears from Narducci’s discovery.

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⁴ Do. p. 53.

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The ultimate Phœnician origin of the Indian alphabets being, as I have already shown¹), admitted on grounds that, except new discoveries of an unexpected kind be made, are tolerably conclusive; it is natural (though not absolutely necessary), to look to a similar foreign source for the elements of the Indian systems of notation. The Phœnician inscriptions supply ample evidence that such a system was in use long before the Indian alphabets could have been borrowed, for that must have happened in comparatively late times, and they also supply proof that the source of this system was that used in Egypt²). The Egyptian hieroglyphic system, in fact, can hardly be said to differ from the Phœnician.

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¹ pp. 4, 7 *etc.*

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² *Cfr.* Z. d. D. M. G. xxi. p. 486.

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The following forms, though collected from inscriptions of different countries and dates, will show the elements of the Phœnician system of numerals: || = 2³); ||| = 3⁴), |||| = 4⁵); 𐤖 or 𐤗 = 10⁶); *H* or 𐤘 = 20⁷). For ‘hundreds’ the sign 𐤙⁸) or 𐤚⁹) is used, and to express the number of ‘hundreds’ dots or strokes are put at the right. Thus 𐤖𐤘𐤘𐤙| = 150¹⁰); ||| 𐤘 𐤘 𐤚 • • = 243¹¹). In the Palmyrene system we find an adaptation of this. The hieratic and demotic numerals (a development and simplification of the hieroglyphic or primitive system) were also certainly used in Egypt before the possibility of any civilization in India. Thus the Egypto-Phœnician system of numerals was in wide use long before we find traces of any such system in India; and there can be no doubt that the Phœnician was the system used by those natives of India who wrote their language in the N. Açoka characters¹²). If, however, we compare the numerals in the S. Açoka inscriptions, it is evident that, though the system there found more or less corresponds with the Egypto-Phœnician, a marked development is, nevertheless, presented in some respects. The circumstances already mentioned (on p. 60) show that this development was going on in the third century B. C.; it consisted not in a general modification of the system, but in the invention of new and simpler symbols for the compounds which were hitherto in use. In the last discovered (and perhaps latest) Açoka inscriptions examples occur which show that about the third century B. C. distinct figures were in use for the units, tens and also for hundreds; the last being qualified by marks on the right to express the number of hundreds. The general similarity of this system to the Egyptian demotic is thus, I think, sufficient to warrant a provisional conclusion that the S. Indian (Açoka) system is derived from it, but developed in India. Of the Egyptian demotic figures for units, those for 1, 2, 3, 4, 5, 6, 7, 8 and 9 have a striking likeness to the corresponding Cave numerals¹.

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³ Inscription of Carthage.

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⁴ Inscription of Umm al Awāmid.

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⁵ Inscription of Eschmunezer.

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⁶ Inscriptions of Eschmunezer and Marseilles.

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⁷ Inscriptions of Marseilles and Umm al Awāmid.

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⁸ Inscription of Marseilles.

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⁹ Inscription of Umm al Awāmid.

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¹⁰ Inscription of Marseilles.

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¹¹ Inscription of Umm al Awāmid.

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¹² E. Thomas in “Numismata Orientalia” pt. i., p. 19.

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¹ M. Barth has never had a doubt that the Cave numerals are of Egyptian origin.

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But it is necessary to notice here a recent theory which might seem, at first sight, to be decisive against this hypothesis, especially as it appears to warrant the conclusion that these numerals are of purely Indian origin. If, however, the real bearing of this discovery be considered, it will, I think, prove rather a remarkable indigenous development, not an independent invention, and is, thus, most in accordance with what we know of the process of adaptation in India, *e. g.* of the Greek astronomy², and the evident traces of adaptation and extension in the S. Açoka alphabet.

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² Albīrūnī (Reinaud, “Mémoire” p. 334) gives a remarkable instance of the Indian tendency in this way: “Les livres des Indiens sont rédigés en vers; les indigènes croient, par là, les rendre plus aisés à retenir dans la mémoire; ils ne recourent aux livres qu’à la dernière extrémité. On les voit même s’attacher à apprendre des vers dont ils ignorent tout à fait le sens. J’ai reconnu, à més dépens, l’inconvenient de cet usage. J’avais fait, pour les indigènes, des extraits du traité d’Euclide et de l’Almageste; j’avais composé un traité de l’Astrolabe à leur intention, afin de les initier aux méthodes des Arabes; mais aussitôt ils mirent ces morceaux en slokas, de manière qu’il était devenu peu facile de s’y reconnaître.” — I have myself seen the Penal Code put into Tamiḻ verse !! On the Indian treatment of foreign names, see my “Aindra Grammarians”, p. 109.

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It has been known for some time³ that the books of Nepal and those of the N. Jains have their pages numbered by a singular series of letters which are ‘e’ (1); ‘dvi’ (2); ‘tri’ (3); kha’ (4); ‘tṛi’ (5); ‘phra’ (6); ‘gra’ (7); ‘hra’ (8); ‘ō’ (9); ‘bṛi’ (10); ‘tha’ (20); ‘la’ (30); ‘su’ or ‘sū’ (100) etc. As primitive customs and usages often linger in out-of-the-way corners like Nepal, it was a very shrewd idea of Paṇḍit Bhagvānlāl Indrajī⁴ to compare these syllable-figures with the Cave numerals, and the resemblance, at once found in many cases, is very striking, but there are most serious difficulties in the way of accepting these Buddhist numerals as a *complete* explanation of the Cave forms. *a*) It must be observed that the Nepal Buddhists and also the Northern Jains have a distinct series of syllables for *all* the units; but the Cave numerals show at once that the figures for the units from one to five, at all events, cannot possibly have been derived from such syllables¹; the figures for *one*, *two* and *three* consist of a corresponding number of strokes, those for *four* and *five* are derived from a cross and a cross with the addition of a curved line. But the figures for the units are of the greatest importance as showing the origin of the system, for they were certainly always in common use, and thus could not readily be changed. Again, I cannot see the least resemblance between the *oldest* forms of the Cave figures for *seven*, *eight*, *nine* and *ten*, and the syllables ‘**gra**’, ‘**hra**’, ‘**o**’ and **bṛi** as used to mark those figures in the Nepal system. The partial resemblance of the figure for *six* to **phra** seems accidental. Where the evidence should be clearest, it thus fails entirely. With still higher numbers the explanation often again fails, for strokes or figures of units are added to the right side of the figures used to express ‘hundred’ and ‘thousand’, and here, again, there is a striking resemblance to the Phœnician system. How by an indigenous system of notation 100×4 could be equal to 400 is very hard to understand. *b*) It also remains to be explained on what principle these syllables were selected. The meaning of ‘**e**’, ‘**dvi**’ and ‘**tri**’ is plain, but the rest baffle all attempts to trace their meaning, and, thus, there is reason to believe that the selection was not made on a consistent principle throughout, as one might expect. *c*) It must, lastly, be noticed that the resemblance to the syllables in question can only be said to begin with later forms of the numeral figures, such as the Gupta and Valabhi; in these it is plain, but it is not so in the earlier forms.

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³ Cowell and Eggeling, “Catalogue of Buddhist Sanskrit MSS.” p. 52.

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⁴ *Indian Antiquary*, vi., p. 48 etc.

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¹ Profr. Kern in *Indian Antiquary*, vi., p. 143.

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If it be kept in mind that the Egypto-Phœnician system of notation was certainly known and used in N. India², and also that the syllabic origin of the Indian figures can only be fully shown in certain cases³ where (*e.g.* 80 and 90) that system had originally no distinct marks but used compound figures; if, again, it be remarked that the resemblance of the numeral figures to the syllables in question—at least so far as the units and the figures for 20, 100, 1000 go—is hardly to be traced in the earlier, but is plainer in the later form of the figures, I think that the natural inference to be drawn is that this proposed explanation of the Indian (Cave) numeral figures by assigning a syllabic origin for them, can only be accepted in some cases (*e.g.* 80 and 90) and that in others (or the majority of cases) it has arisen from a fancied resemblance found between the Cave numerals and certain syllables, but which, so far as their origin goes, have really nothing in common.

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² Above p. 64.

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³ It may be remarked that of the two types we find in the Cave numerals, it is only in one (the Western) that a full resemblance to the syllables suggested can be found, as even Dr. Bühler (the chief advocate of this theory) admits in fact: “Three New Edicts of Açoka”, pp. 23-5.

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The explanation is a remarkable one, but, thus, very limited in its results. Except, however, new and ample materials (from the early centuries B. C. and A. D.) be yet found for deciding the question, every explanation of the ultimate origin of the S. Indian system of notation must be regarded as merely a provisional hypothesis[^1).

§ 1. The Modification of the 'Cave' Numeral Figures Found in the Veṅgi and Cēra Inscriptions

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The Cave Numerals, given in Plate xxiii., are taken from those which occur in the inscriptions of the Western Caves as far as the second line is concerned; the upper are from the Mathura inscriptions[^2). The two inscriptions of the Veṅgi dynasty (as I have termed it already) which preceded the Cālukyas, and therefore must be earlier than the seventh century A. D., have the plates numbered. In one, numerals occur up to *three*, and in the other (which is given in Plate xxiv.) up to *four*; these are collected in Plate xxiii. The Pallava inscriptions (which are a little later) supply the figures for *five* and *six*.

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The horizontal strokes of the Cave numerals are here semi-circular, and the figure for *four* is also of a more cursive form; the figures for *five* and *six* are also modified.

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Much the same numeral figures appear to have been in use in the Cera kingdom at the end of the fifth century A. D. In the Mercara Plates (ii. line 9) ∞ “sahasranāḍu” occurs[^3). This is left unexplained by those who have attempted this inscription, but the figure is evidently a slight variation of the Cave numeral 10, and the words thus should be read “daçasahasranāḍu”; the ‘ten-thousand’ being a division of the country, and probably referring to the tribute paid by it.

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I have not met with any other examples of this system of numerals in Southern India.

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[^1): In considering this question it will be well to recollect what Comte says: “The fortunate custom of hieroglyphic writing....... led to the permanent adoption of hieroglyphics in the case of numbers.” He also remarks that the decimal notation (with value by position) could not have come from alphabetic writing. (“Pos. Polity” English tr. iii., 182.) Now in India there is no trace of indigenous hieroglyphs. [^2): *Bombay Journal*, viii., pp. 225-232; and *Journal of the R. A. S.* New Series, V. pp. 182 ffg. [^3): A good facsimile of these very important plates is given in the first volume of the *Indian Antiquary*. The explanation, however, needs much amendment.

§ 2. The Tamil Numeral Figures

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The figures used in this system are given in Plate xxiii. from inscriptions of about 1520 and a ms. at Tanjore which belongs probably to the end of the sixteenth or beginning of the seventeenth century; as Tamiḷ mss. (except the very recent ones) are all undated, and these figures rarely occur in inscriptions earlier than the sixteenth century, it is difficult to procure a complete series of an ascertained date. This is, however, of little importance; for the earliest examples known (about 1400 A. D.) are precisely of the same form as those still in use.

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These figures are remarkable as forming the stage of development between the W. Cave numerals and the modern systems, and are, therefore, relics of a system that became more or less obsolete in the sixth century A. D.¹; we find here separate figures for *ten*, *hundred* and *thousand* nearly identical with the W. Cave forms; but the figures for *twenty* etc. are rejected, and tens, hundreds or thousands are expressed by prefixing the sign for the units to the left side of the figure representing the order. The use of the cipher and value of position are Grantha (or Brahmanical), and till lately have been but little used, though Sanskrit mss. are almost invariably numbered in this way.

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¹ The Kuṟaḷ (xi., 2) mentions acquaintance *with numbers* (eṇṇa) and *letters as being like eyes to men*. This is of about 850 A. D.

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The figures used to express fractions are peculiar to the Tamiḷ people, and there are many others in use besides those which I have given, and which I have chiefly taken from the first edition of Beschi's Koḍun-Tamiḷ Grammar (p. 149). They are derived, no doubt, from initials of corresponding words, which abbreviations are also combined in some cases; the invention must be attributed to the Tamiḷ traders of no very remote period².

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² In the older inscriptions (at Tanjore *e. g.*) all numbers and fractions which occur, are written at full length.

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The Tamiḷ numeral figures are obviously cursive forms of the Cave numerals modified by the prevailing practice of writing on palm-leaves with a style, a practice which renders necessary curved rather than straight lines, as the last, when with the grain or course of the fibres of the leaf, are nearly invisible.

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I have not been able to find any traces of distinct Vaṭṭeḻuttu numerals.

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The Malayāḷam numerals (which I have given in Plate xxiii.) are those in actual use. Their history is quite uncertain, as there are very few, if any, examples of them older than the middle of the last century, mss. being numbered most generally with letters. They are evidently derived from the same source as the rest, and are nearest to the Tamil figures, but include the cipher. The Malayāḷam way of expressing fractions is the same as we find in the Telugu and Canarese countries, and is, therefore, North-Indian.

§ 3. The Telugu-Canarese Numeral Figures

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These figures appear in common use about 1300 A. D.¹ with value by position and also the cipher, which is always represented in S. Indian documents by a small circle. In Northern India a dot also appears with this signification, but the necessity of writing on palm-leaves has, in S. India, led to the adoption of the circular form as alone perfectly distinct. The earliest specimens of these numerals that I have met with occur on the outer rims of the plates of a çāsana of 1087 A. D., already mentioned (p. 21 *n.*) as E. These do not go beyond nine, but the cipher is plain in an inscription of much the same date.

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¹ If Sir W. Elliot's collection of transcripts of inscriptions in the neighbourhood of the Krishṇā and Godāvari can be trusted, the notation of dates by these numerals was not uncommon in the eleventh century; but I am inclined to think that this is not the case, and that the copyist has often put the figures for words written at full length in the original. The oldest inscription with a date in figures in Java appears to be that given in pl. ii. of Dr. Cohen-Stuart's “Kawi Oorkonden”; it is of the 9th century, and is, therefore, strangely enough, older than any S. Indian document with a date in figures.

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The Telugu-Canarese numerals (as given in Plate xxiii. from a Haḷa-kannaḍa ᴍꜱ. of 1428 A. D.) are almost identical in all the inscriptions across the peninsula, and remained the same till quite recently. In the Telugu inscriptions I have, however, observed, in some cases, a slight difference in the form of the figure 5, which sometimes wants the middle connecting stroke. The figure 3 is generally perpendicular in the Telugu inscriptions.

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The Telugu-Canarese system of fractions is, like the Tamil, based on a division of the unit into sixteen parts; they are marked by the N. Indian system, and this appears to be of recent introduction.

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A comparison of the numeral figures in Plate xxiii. will conclusively show that they are all more or less cursive modifications of the Western Cave numerals. As the Cave numerals are from Western and Northern India, and present a number of distinct types, there is no real difficulty about some which present variations of form, for the perfectly evident origin of 1, 2, 3, 4 and 7 quite justifies the conclusion that the smaller number, of which the origin is less obvious, do in fact come from the same source². The origin of the S. Indian numeral figures is, thus, the same as that of the S. Indian alphabets.

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² The Gobar is from Woepcke's “Mémoire” p. 49; the Nāgari is from Prinsep's “Essays” as collected by Mr. Thomas.

§ 4. The Javanese Numeral Figures

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One of the most interesting features of the Kawi and Sanskrit inscriptions of Java is the complete information they give about the numerals.

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From the 9th century A. D. onwards we find the decimal system of notation in full use[^1).

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The earliest certain specimen[^2) is of 841 A. D., and it is easy to see that these figures are a slight modification of those in the Vengi and Pallava inscriptions, such as would occur in the course of some four hundred years which is the difference of date between the two. After this period the progress of change is rapid, and by the 14th century it had gone so far that the figures used are wholly unlike those which are their original source. This fact is remarkable as, except in the case of a few letters, the Kawi character underwent but little change in form for a long while. In later times the modification has been very rapid.

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It is necessary to remark that as the plates which compose çāsanas are very rarely numbered, it is very difficult to get complete series of the numeral figures used in earlier times. It cannot, however, be doubted that they were in much more common use, than the rarity of their occurrence in old documents would lead an observer to infer.

Note: The Different Methods of Marking Dates Used in South-India

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The numeral figures are only used in comparatively modern inscriptions; in the older ones and also in many modern ones the numbers are commonly expressed by words or letters. The eras and cycles to which the dates are referred also present considerable difficulty, owing to the variations found in different places.

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[^1): *Cfr.* plates i., ii., v., vii., ix., xi., xiv., xv. and xxii. of Cohen-Stuart's “Kawi Oorkonden.” Z. d. D. M. G. xviii., pl. 495 etc. [^2): There is another of a few years earlier, but it is very illegible, and, therefore, uncertain.

A. The Eras

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a. The Kaliyuga

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The usually received date of the Kaliyuga is the March equinox of 3102 B. C.

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It was known in the fifth century A. D.¹, but has never become very general in inscriptions, and is now, in S. India, chiefly used in Malabar for the fanciful way of marking dates by a sentence. In most cases I have seen, the number of *days* and not of *years* is mentioned². I believe that the use of this date is unusual, except in comparatively modern times, and is a ground for doubting the authenticity of any but modern inscriptions. It is the base of the ‘ahargaṇa’ system of the Indian astrologers, and the use of it to indicate dates is, almost certainly, due to these men who would naturally be employed to calculate the dates of grants.

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¹ By Āryabhaṭa (*e. g.* iii., 10). Albīrūnī mentions it. Reinaud, “Fragments”, p. 136.

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² Warren’s “Kala Saṅkalita” (p. 18) states that in S. India it is usual to date documents in both the Kali and Çaka year. This is contrary to my experience, so far as documents before 1500 are concerned.

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b. The Çaka Era

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This era is now usually supposed to date from the birth³ of a mythical Hindu sovereign called Çālivāhana, who defeated the Çakas, and began Monday 14th March 78 A. D. (Julian style). The account of the origin of this era has apparently been repeatedly modified to suit current ideas. In the earlier inscriptions it is usually called ‘Çakavarsha’, ‘Çakasamvatsara’ or ‘Çakanṛipakāla’; about the tenth century it is termed the year of the ‘Çakarāja,’ ‘Çakādhipa’ or ‘Çakadeva,’ and still later it is termed ‘Çālivāhanaçaka’ or ‘Çālivāhanaçakābda’.

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³ Some say: from the consecration.

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Albīrūnī (A. D. 1031) speaks of this era as one in use by the astrologers⁴; and as they had a great deal to do with royal grants by determining the auspicious time for making them, it is easy to see how this became the most usual way of marking the dates of inscriptions. But it is certain that this era was quite unsettled and comparatively little used before the tenth century. The earliest authentic inscriptions in which it occurs belong to the end of the fifth century A. D., but it is first mentioned by Varāha Mihira, an astronomer who lived in the sixth century A. D.; and he makes the commencement of it coincide with Kali year 3179. The great popularity in all parts of India of this author's works is probably the reason why this is now the recognized computation, but it has been adopted since the tenth century. Up to that date and even later, there are inscriptions with dates by the Çaka as well as other methods, (*e. g.* the Bṛihaspati cycle) which show a variation of two or three years, more or less, from the usual computation. Albīrūnī (A. D. 1031) mentions that the Çaka year then commenced 135 years after Vikramāditya; this is the received opinion, and from that century Çaka dates may be computed with certainty in the ordinary way. Before that period they must be considered as more or less uncertain.

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⁴ “L’ère de Saca, nommée pas les Indiens Sacakâla, est postérieure à celle de Vikramaditya de 135 ans. Saca est le nom d’un prince qui a régné sur les contrées situées entre l’Indus et la mer. Sa résidence était placée au centre de l’empire, dans la contrée nommée Āryāvartta. Les Indiens le font naître dans une classe autre que celle des sakyas; quelques-uns prétendent qu’il était soudra et originaire de la ville de Mansoura. Il y en a même qui disent qu’il n’était pas de la race indienne, et qu’il tirait son origine de régions occidentales. Les peuples eurent beaucoup à souffrir de son despotisme, jusqu’à ce qu’il leur vînt du secours de l’Orient. Vikramāditya marcha contre lui, mit son armée en déroute et le tua sur le territoire de Korour, situé entre Moultan et le château de Louny. Cette époque devint célèbre à cause de la joie que les peuples ressentirent de la mort de Saca, et on la choisit pour ère, principalement chez les astronomes”. Tr. by Abbé Reinaud, “Mémoire”, pp. 79-80, and in “Fragments Arabes et Persans inédits relatifs à l’Inde”, pp. 140-141, 145.

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The Çaka year seems to have been originally introduced by the Digambara Jains, but though the inscriptions prove that their computation of it was the same as the brahmanical, the account they give of it differs from the ordinary one. The Trilokasāra says: Paṇachassayavassaṁ paṇamāsajudaṁ gamiya Vīranibbuïdo | Sagarājo; to Kakki caduṇavatiyamahiyasagamāsaṁ || 848 || C. Çrī-Vīranāthanivṛitteḥ sakāçāt pañcottara-shaṭchatavarshāṇi pañcamāsayutāni gatvā paçcāt Vikramāṅkaçakarājo 'jāyata | tata upari caturṇavatyuttaratriçatavarshāṇi saptamāsādhikāni gatvā paçcāt Kalky ajāyata || Now the death of Vīranātha (or Mahāvīra) the last of the Tīrthaṅkaras is put at 388 B. C.[^1); then, according to the above, the Çaka era would begin in 239 A. D., but this is impossible, so the era of Mahāvīra must be put at 527 B. C., and this again differs from the era mentioned by Prinsep as current in the North of India—512 before Vikramāditya or 569 B. C.[^2) The Javanese Çaka era is 74 A. D., that of Bali 80 A. D. Friederich ("Over Inscriptiën van Java en Sumatra" p. 78) says that only the Çaka era has been found in use in the archipelago. From these details some notion may be formed of the excessive uncertainty of Indian chronological data before the early centuries A. D. The more exact they appear to be the more suspicious they are. It is not too much to say that a tolerably exact chronology is only possible after the tenth century, and then by the aid of inscriptions only[^3).

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It is necessary to remark that the Çaka year is reckoned by either including the current year (the most usual practice in India), or by excluding the current year and

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[^1): According to the Çatruñjaya-Māhātmya. Bühler ("Three New Edicts of Açoka", p. 21) rejects this work as a forgery, and accepts the date 526-7 B. C. [^2): "Useful Tables" p. 166 in Prinsep's "*Essays*" by Thomas, Vol. II. [^3): The rough equation for converting this era into the Christian date is: +78¼. The beginning of the year being at the March equinox; if the Çaka '*atīta*' year be mentioned, the equation is: +79¼. For more exact reckoning, a long calculation is necessary.

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giving only the number of complete years that have elapsed, in which case the word atīta is used; this last is not usual in Indian (except the W. Cālukya) inscriptions, but is nearly universal in those of Java.

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c. The Vikramāditya Era

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The passion for systematizing and thus falsifying even history in accordance with the popular astrological and religious notions of the day, has, it is evident from the above, led to repeated alterations in the dates assigned to real or fictitious events in Indian history. The era of Vikramāditya is apparently one result of this folly. It is all but unknown in S. India (except in the Deccan), though under the name of ‘*saṃvat*’ is the one most commonly used in the North. It is said to begin 57 years B. C.¹ It is used by the Çvetāmbara Jains (of the N. of India).

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¹ The equation is: +56¾. It is mentioned by Albirūnī (Reinaud, “Fragments”, p. 139).

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d. The Kollam, Koḷambam (or Quilon) Era

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This is usually called a cycle², but it is in reality an era; it began in September 824 A. D. It is supposed to commemorate the founding of Kollam (Quilon), and is only used in the S. Tamiḷ country and Travancore³.

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² “Cycle of Paraçurāma”—Prinsep.

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³ The equation is: +824¾.

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e. Cycle of Bṛihaspati

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⁴ The first account, by an European, of this Cycle, and a very good one too, is in the “Open-Deure” (1651) of Rogerius who lived from 1631 to 1641 at Pulicat as Dutch chaplain. (See pp. 58-9.) *As. Res.* iii. “On the Cycle of 60 years” by Davis. Sūryasiddhānta by Burgess, (New Haven, 1860) p. 35,—a list of names (fr. Davis) is given on p. 36. This list is not to be found in the Sanskrit treatises, but is supposed to be generally known; for this reason it is impossible to amend some of the names which are clearly corrupt.

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Each year in this cycle has a name, and in the inscriptions this is coupled with the Çaka year or year of the king's reign. The earliest examples to be met with in S. India in which the cyclic years occur are of about the tenth century. The names are as follows:

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¹ According to Mr. C. P. Brown the order is sometimes: *Jaya*, *Vijaya*.

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² This list is compiled from Col. Warren's "Kala Saṅkalita", Mr. C. P. Brown's "Cyclic Tables", inscriptions, and the practice of the people of S. India. I am not aware that any old list exists.

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This cycle is originally founded on a practice of reckoning time by the revolutions of Jupiter (Bṛihaspati), but there is no record of its correct use; the present practice of erroneously reckoning sixty *solar* years as equal to five revolutions of the planet has always, it appears, prevailed as far back as reference to this method can be found. Though this cycle is in common use everywhere in the South, the names are often much varied, especially by the Jains³. It is not improbable that this system is an adaptation with Sanskrit names of an old way of reckoning time originally current in S. India; it is mentioned by Albīrūnī¹ in the eleventh century, but his reference to it is commonly understood to mean that it was of recent introduction in the North and West of India².

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³ The Tamiḻ names are merely corrupt forms of the Sanskrit. For them, see Beschi's Koḍun-Tamiḻ Grammar. To find the year A. D. corresponding to the first of the cycle it is not necessary, here, to do more than remark that the last must always end with 7, and that the 11th, 21st, 31st, 41st, 51st years of the cycle (as used in S. India) must also correspond with A. D. years ending with 7. Thus the first years of cycles would correspond with A. D. 67, 127, 187, 247, 307, 367, 427, 487, 547, 607, 667, 727, 787, 847, 907, 967 etc. The corresponding years Çaka and A. D. are given by Mr. Brown down to 1857.

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¹ Reinaud, "Fragments" p. 140 *n.*, states that Albīrūnī devotes an entire chapter to this cycle.

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² Do: *u. s.* An inference as to the recent date of the origin of this cycle (viz., 959 A. D.) is drawn by the Abbé Reinaud, but it is certainly wrong. The Bṛihaspati-cycle is referred to, *e. g.*, by Āryabhata (iii., 10) who lived at the end of the 5th century A. D. and the years of it occur commonly in inscriptions of that century. It does not appear in use in the Kawi inscriptions.

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This cycle, as used in North and South India, differs not in the names or order of the names but in the period at which the first year comes. In S. India the present year (1874) is Bhāva or the eighth of the cycle. This difference is owing to the practice which obtains in S. India and Tibet of considering the years of the cycle as identical in duration with the ordinary luni-solar year.

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f. Other Eras but Little Used

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Some of the Cālukyas attempted to set up local eras, but these dates occur in comparatively few and unimportant inscriptions, and are too uncertain to be worth mentioning here.

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The South-Indian Coḷa and Pāṇḍya kings appear to mention the year of their reign most generally, and the second also, but rarely, the Quilon era. The task of establishing the succession of these dynasties and the dates is thus likely to prove very formidable; there is, however, some foundation in Marco Polo's mention of Sundara Pāṇḍya as the king of the South in his time (13th century), and also in the synchronism between the Coḷa king Kulottuṅga and the Cālukya Āhavamalla as established by Sir W. Elliot³.

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³ *Madras Journal*, xiii., pt. 2, p. 40. See above p. 19 *n.* and p. 40 *n.*

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The explanation of the date in the grant to the Cochin Israelites is not as yet certain. The term is: "Yāṇḍu iraṇḍām āṇḍaikk 'ĕdir muppattāṟām āṇḍu"—*i. e.* (literally) "the year opposite the second year, the thirty-sixth year." Ellis explained it⁴ by the thirty-sixth year of the third (? second) cycle, but it is impossible to reconcile it with the Quilon era, and it appears to me here to mean the thirty-sixth year (of the king's life) opposite to (or corresponding with) the second year (of his reign or of the cycle). Similar dates occur in the Tamiḷ inscriptions, and the meaning of mēl viz., 'equivalent to', is beyond doubt. Thus in an inscription of A. D. 1532 (Piḷḷaiyār temple at Tanjore)

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⁴ Do. pp. 3 and 10. Dr. Gundert (do. pt. i., p. 137) doubts the meaning of ĕdir, and the usual translation is rather of 'mēl' than of 'ĕdir', for it is by no means certain that the two have the same meaning. Dr. Caldwell (*Comp. Grammar*, p. 60 *n.*) takes it to mean the year of the cycle of sixty to which the year of the king's reign answers.

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we find: "Çakābdam 1454 iðin mēl nandanasamvatsaram", and in fact, ç. 1454 corresponds with the cycle year Nandana.

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It was a practice in the Tamil inscriptions to note the day of the year by a number, *e.g.* "the 26th year, 310th day" (Tanjore inscription, iv.), or even to count the days of a king's reign¹. In this, as in so many other details, the Tamil inscriptions are unlike those of the North, and form, unfortunately for the chance of obtaining exact dates, a class by themselves.

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Linked texts (10)

Aṣṭādhyāyī — Pāṇini, Sanskrit
Tirukkuṟaḷ — Tiruvaḷḷuvar, Tamil
Sūryasiddhānta — Maya, Sanskrit
Chandaḥsūtra — Piṅgala, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Ārcajyotiṣa — Lagadha, Sanskrit
Tanjore — inscription (SII 4.1289) — Tamil
Rûpnâth
Shahbaz Garhi
Tiloyasāra — Nemicandra Siddhāntacakravartin, Śaurasenī Prakrit